00:01
Okay, so we've got that everlast batteries, they say, have a normal distribution with mean 40 and standard deviation 4.
00:12
Then a new chemical composition is introduced and we want to see if this has affected the lifetime of the batteries.
00:19
So the null hypothesis is going to be that the new mean life is still 40, that it hasn't affected it, and the alternative hypothesis is just going to be that it's not equal to 40, i .e.
00:30
That it has affected it.
00:33
And to test this we take a sample of 100 batteries and the mean lifetime is 39 .1.
00:41
Question 1 asks us to compute the test statistic and p -value of this test.
00:45
We have a sample size of over 30 and we know the population standard deviation, so we're going to use the z -test and the z -test statistic is given by x -bar minus mu over sigma over root n, so that in this case is 39 .1 minus 40 divided by 4 over root 100.
01:04
And that gives us a z -test statistic of minus 2 .25 and the p -value associated to this statistic you can find in your p -tables, the probability of it being less than 2 .2 minus 2 .25 is 0 .0122.
01:27
And then because we're doing a two -tailed test we want the probability of it being bigger than plus 2 .25 as well, so we just times this by 2 and we get 0 .0244.
01:42
Part 2 says at what significance levels would you reject it, particularly at 0 .05 and at 0 .01? so our p -value is less than 0 .05 but bigger than 0 .01, so at 0 .05 we would reject h0 and at 0 .01 we would fail to reject h0.
02:08
Part 3 asks the critical region for the test at 0 .05 significance level and at 0 .01.
02:15
So at 0 .05 what it means by the critical region is that we want the z -values which cut off the outside 5%, so that means that 0 .025 are outside, and so the z -rejection region is that z is bigger than 1 .96 or z is less than minus 1 .96 and this corresponds to a sample mean of x bar being bigger than 40 .784 and less than 39 .216.
03:12
For part b we do the same but with a 0 .01 significance level, so the only thing that changes is that now we want the z -values that bound the outside 1 % and you can find that that is plus and minus 2 .57...